Gravitationally trapped photons leads to quantized space-time

[johne1618]

Hi,

I was wondering what happens if one has a photon that is so energetic that it gets trapped in its own gravitational field.

Imagine a mass m orbitting a large mass M in a circular orbit with radius r and with transverse velocity v. By equating the gravitational force on m with its centripetal acceleration we have:

G M m / r^2 = m v^2 / r

Now let velocity v approach the speed of light c. The mass m will be boosted by a Lorentz factor but I think the equation should still hold so that we have:

G M m / r^2 = m c^2 / r

Thus cancelling m we have:

M = (c^2 / G) r

or in terms of Energy E = M c^2 we have

E = (c^4 / G) r (*)

This is the energy that a photon must have to get trapped in its own gravitational field so
that it orbits in a circle with radius r. The quantitiy c^4 / G is also the string tension so maybe this is also a model of a string.

Now let us suppose that the photon has angular momentum hbar. Then we have

r * p = hbar

where p is the linear momentum of the photon.

Now we know that p = E / c for photons so that we have:

r * E / c = hbar

E = hbar * c / r

E = h * c / 2.pi.r

E = h / t (**)

where t is the time period of the photon orbit.

If we substitute E=h/t into equation (*) we get:

h / t = (c^4 / G) * r

Rearranging we get:

r * t = h G / c^4

Thus we find that the spacetime area occupied by this bound photon, r * t, is quantized.

I was wondering if this is similar to the result that the world sheet area swept out by strings is quantized.

 

[robert2734]

>I was wondering what happens if one has a photon that is so energetic that it gets trapped in its own gravitational field.

That's why we know there is New Physics at the Planck level. The whole world wants to know what happens when a photon's wavelength is smaller than it's scharzchild radius. i. e. at the Planck energy/length.

 

[freejwho]

In addition, the assumption
"'Now let velocity v approach the speed of light c. The mass m will be boosted by a Lorentz factor but I think the equation should still hold so that we have:

G M m / r^2 = m c^2 / r"
.. is not suitable. Transition to special relativity is needed.

 

[robert2734]

The schwartzchild radius r=2MG/c^2 is valid even when general relativity is considered.

 

[Nabeshin]

A quick example of why mindlessly attempting to apply Newtonian theory to both light and general relativity is inadequate:
From your derviation you would no doubt agree for a circular orbit.
$$v=\sqrt{\frac{GM}{r}}$$
Suppose we want to find the radius of a photon's orbit, so v=c, we get
$$r=\frac{GM}{c^2}=\frac{r_s}{2}$$
Which is inside the event horizon of the black hole! Clearly, something went wrong along the way.

I'm not going to poke holes in everything you've written, but you should realize that your treating GR and photons so lightly is not appropriate.